What is the limit as t approaches 0 of 1/t-1/(e^t-1) ?

The limit as t approaches 0 of 1/t-1/(e^t-1) is 1/2

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limit as t approaches 0 of 1/t-1/(e^t-1)

t \to 0 lim (\frac{1}{t} - \frac{1}{e ^{t} - 1})

\frac{1}{2}

Decimal Notation

0.5

Solution steps

t \to 0 lim (\frac{1}{t} - \frac{1}{e ^{t} - 1})

Simplify

\frac{1}{t} - \frac{1}{e ^{t} - 1} : \frac{e ^{t} - 1 - t}{t ( e ^{t} - 1 )}

= t \to 0 lim (\frac{e ^{t} - 1 - t}{t ( e ^{t} - 1 )})

Apply L'Hopital's Rule

: t \to 0 lim (\frac{e ^{t} - 1}{e ^{t} - 1 + e ^{t} t})

= t \to 0 lim (\frac{e ^{t} - 1}{e ^{t} - 1 + e ^{t} t})

Apply L'Hopital's Rule

: t \to 0 lim (\frac{e ^{t}}{e ^{t} t + 2 e ^{t}})

= t \to 0 lim (\frac{e ^{t}}{e ^{t} t + 2 e ^{t}})

Plug in the value

t = 0

= \frac{e ^{0}}{e ^{0} \cdot 0 + 2 e ^{0}}

Simplify

= \frac{1}{2}

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What is the limit as t approaches 0 of 1/t-1/(e^t-1) ?
The limit as t approaches 0 of 1/t-1/(e^t-1) is 1/2